Optimal. Leaf size=79 \[ \frac {2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {2 \sqrt {5 x+3}}{49 \sqrt {1-2 x}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {2 \sqrt {5 x+3}}{49 \sqrt {1-2 x}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\frac {2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{7} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=-\frac {2 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac {1}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac {2}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {2 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {2 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.90 \[ \frac {2 \left (7 \sqrt {5 x+3} (41 x+18)+3 \sqrt {7-14 x} (2 x-1) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{1029 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 86, normalized size = 1.09 \[ -\frac {3 \, \sqrt {7} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (41 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1029 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 113, normalized size = 1.43 \[ \frac {1}{3430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {2 \, {\left (41 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3675 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 154, normalized size = 1.95 \[ \frac {\left (12 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-12 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+574 \sqrt {-10 x^{2}-x +3}\, x +3 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+252 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1029 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 104, normalized size = 1.32 \[ \frac {1}{343} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {205 \, x}{147 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x^{2}}{6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {37}{588 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1385 \, x}{84 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {67}{28 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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